140 (1991)
On D -dimension of metrizable spaces*
by
Wojciech O l s z e w s k i (Warszawa)
Abstract. For every cardinal τ and every ordinal α, we construct a metrizable space Mα(τ ) and a strongly countable-dimensional compact space Zα(τ ) of weight τ such that D(Mα(τ )) ≤ α, D(Zα(τ )) ≤ α and each metrizable space X of weight τ such that D(X) ≤ α is homeomorphic to a subspace of Mα(τ ) and to a subspace of Zα+1(τ ).
1. Introduction. Our notation and terminology follow [1] and [2]. By dimension we mean the covering dimension, by space a normal space, and by mapping a continuous mapping. We use the habitual convention that an ordinal α is the set of all ordinals less than α.
The symbol |A| denotes the cardinality of the set A, the symbols N and I the set of non-negative integers and the closed unit interval, respectively;
i, j, k, l, m, n denote natural numbers, α, β, γ, δ, ξ, η ordinals, λ a limit ordinal, τ an infinite cardinal, ℵ0 the smallest infinite cardinal, and %, σ metrics.
Let X be a space. We put D(X) = −1 whenever X = ∅. If X 6= ∅ and α = λ + n, then D(X) ≤ α whenever there exists a closed covering {Aβ : β ≤ λ} of X consisting of finite-dimensional subsets such that:
(1.1) for every δ ≤ λ, the set S{Aβ : δ ≤ β ≤ λ} is closed,
(1.2) for every x ∈ X, there exists a greatest ordinal β ≤ λ such that x ∈ Aβ,
(1.3) dim Aλ≤ n.
If, apart from that, Aλ= ∅, then we write D(X) <≤ λ. If there exists an α such that D(X) ≤ α, then D(X) is the smallest such α; in the opposite case, we set D(X) = ∆, where ∆ > α and ∆ + α = ∆ for every ordinal α.
The existence of a closed covering {Aβ : β ≤ λ} of X by finite-dimen- sional subsets satisfying (1.1)–(1.3) implies the existence of such a covering
* Supported by Polish scientific grant RP.I.10.
{A0β : β < λ} satisfying (1.1)–(1.3) and
(1.4) if β = γ + m < λ, where γ is 0 or a limit ordinal, then dim A0β ≤ m.
Indeed, it suffices to put
A0γ+m = Aγ+l if m = l + dim Aγ+1+ . . . + dim Aγ+l,
∅ for the remaining m ∈ N for every γ which is either 0 or a limit ordinal less than λ.
The ordinal number D(X) is called the D-dimension of X and was in- troduced by D. W. Henderson (see [4]). If X is a space of weight τ , then
|D(X)| ≤ τ (see [4], Theorem 10).
A space is called strongly countable-dimensional if it is the union of a countable family of finite-dimensional closed subsets. One readily checks that a space X is strongly countable-dimensional iff X =S{Xn : n ∈ N }, where Xn is a closed subspace of X and dim Xn≤ n for every n ∈ N .
In [10] a class of small spaces was defined and it was observed that in the realm of strongly hereditarily normal (in particular, metrizable) spaces this class coincides with the class of spaces X such that D(X) < ∆; thus it follows from Theorems 3.2 and 3.8, and Corollary 3.4 in [10] that if X is a metrizable space, then D(X) < ∆ iff X has a strongly countable-dimensional completion (see also [6]).
L. Luxemburg showed (see [8], Theorem 1.3) that
(1.5) for every ordinal α such that |α| ≤ ℵ0, there exists a universal space for metrizable spaces X of weight ℵ0 such that D(X) ≤ α.
It is also known (see Conjecture in [5], Theorem 2 in [6], Theorem 8.1 in [7]) that
(1.6) every metrizable space X of weight ℵ0 has a strongly countable- dimensional compactification Z of weight ℵ0 such that D(Z) ≤ D(X) + 1.
An analogous compactification theorem for metrizable spaces of arbitrar- ily large weight, i.e., (1.6) with ℵ0 replaced by an arbitrary τ , follows from the results announced in I. M. Kozlovski˘ı’s paper [6]; their proofs, however, have never been published by the author. This general compactification theorem seems to be particularly valuable in view of the following corollary (see [6], Corollary):
(1.7) every strongly countable-dimensional completely metrizable space of weight τ has a strongly countable-dimensional compactification of weight τ .
The aim of this paper is to generalize (1.5) to metrizable spaces of ar- bitrarily large weight. Simultaneously, we give a proof of I. M. Kozlovski˘ı’s
generalization of (1.6); in fact, we prove a little more—namely, the exis- tence of a common such compactification for all spaces with fixed weight and D-dimension.
We denote by J (τ ) the hedgehog of spininess τ , by d its standard metric (see [1], Example 4.1.5), and by [J (τ )]ℵ0 its countable Cartesian power. The symbol O will often appear in our considerations; it will denote distinguished points of various spaces. In particular, we denote by O the “origin” of J (τ ) and the point of [J (τ )]ℵ0 with all coordinates equal to O.
Every ordinal α > 0 which is not the sum of two ordinals smaller than α is called a prime component. For every ordinal α > 0, there is a unique representation α = α1+ . . . + αk+ αk+1, where αi> 0 is a prime component and αi+1 ≤ αi for i ≤ k, and αk+1 is finite. Let α = α1+ . . . + αk+ αk+1
and β = β1 + . . . + βl + βl+1 be such representations of α and β, and let γ1, γ2, . . . , γk+l+2 be the elements of {α1, . . . , αk, αk+1, β1, . . . , βl, βl+1} written in decreasing order. Then the natural sum α ⊕ β is defined by α ⊕ β = γ1+ . . . + γk+l + (γk+l+1 + γk+l+2). For every ordinal δ, there are only finitely many solutions α, β of the equation α ⊕ β = δ, and if α ⊕ β > γ for some ordinals α, β, γ, then there exist ξ ≤ α and η ≤ β such that ξ ⊕ η = γ. Detailed information about prime components and natural sums can be found in [11] (Chapter XIV.6, 28), and [3] (Chapter IV.14).
The symbol Kn(τ ) denotes the n-dimensional universal Nagata space, i.e., the subspace
{(x0, x1, . . .) ∈ [J (τ )]ℵ0:
|{xi: d(O, xi) is a positive rational number }| ≤ n}
of [J (τ )]ℵ0. If m ≤ n, then Km(τ ) ⊆ Kn(τ ).
Suppose we are given spaces X, Y , a subset A ⊆ X, and a mapping f : X → Y . We say that f separates points of A and closed sets in X if the following condition is satisfied:
(1.8) if x ∈ A and x 6∈ B = cl B ⊆ X, then f (x) 6∈ cl f (B) ⊆ Y .
2. The lemmas. In this section, we shall formulate three lemmas. The second one is a consequence of a theorem announced in [6] (see Theorem 1), but we will quote [9], where the proof of this theorem can be found.
2.1. Lemma. Let X be a metrizable space. For every open covering {Vm : m ∈ N } of X satisfying Vm ⊆ Vm+1 for all m ∈ N , there exists an open shrinking {Wm: m ∈ N } satisfying Wm∩Wl= ∅ whenever |m−l| > 1.
P r o o f. For every m ∈ N , take a function fm : X → I such that
fm−1(0) = X − Vm, and put f (x) =
∞
X
k=0
1
2k+2fk(x) for x ∈ X.
Let Wm = f−1((m+31 ,m+11 )) for m ∈ N . The family {Wm : m ∈ N } has the required properties (cf. the proof of Theorem 3.1 in [9]).
2.2. R e m a r k. In the hypothesis of Theorem 2.12 in [9], we can require that g(F ) is a closed subset of the space g(X). This follows immediately from the proof presented there (see also [6], Theorem 1).
If under the assumptions of Corollary 2.5 in [9], we additionally fix an m- dimensional closed subspace A of X, then in the hypothesis, we can require that cl h(A) ⊆ Km(τ ) for every h ∈ P. This follows from the proof presented in [9], where Corollary 2.7 instead of Theorem 2.1 should be applied. Thus for every n-dimensional metrizable space X of weight not larger than τ , every m-dimensional closed subspace A ⊆ X, and every x0 ∈ X, there exists an embedding h : X → Kn(τ ) such that h(A) ⊆ Km(τ ) and h(x0) = O.
2.3. Lemma. Let X be a metrizable space of weight τ , and E and F its closed subsets. If dim(F − E) ≤ n, then there exists a mapping f : X → Kn+1(τ ) separating points of F − E and closed sets in X such that : (2.1) f (F ∪ E) is a closed subset of f (X),
(2.2) f (F ∪ E) ⊆ Kn(τ ) ⊆ Kn+1(τ ) and f−1(O) = E.
P r o o f. By Lemma 2.9 in [9], there exist a metrizable space X0of weight not larger than τ , a point x0 ∈ X0 and a continuous mapping q : X → X0 such that q−1(x0) = E and q|X − E is a homeomorphic embedding onto X0 − {x0}. Let F0 = q(F ) ∪ {x0}; then F0 is a closed subset of X0 and dim F0≤ n.
By Theorem 2.12 in [9], there exist an (n + 1)-dimensional space Z of weight not larger than τ and a continuous mapping g : X0→ Z separating points of F0 and closed subsets in X0; since x0 ∈ F0, we have g−1(g(x0)) = {x0}, and by the first part of Remark 2.2, we can assume that g(F0) is closed in g(X0). By the second part of Remark 2.2, there exists an embedding h : g(X0) → Kn+1(τ ) such that h(g(x0)) = O and h(g(F0)) ⊆ Kn(τ ). Then the mapping f = h ◦ (g X0) ◦ q has the required properties.
2.4. Lemma. Let X, Y be compact or metrizable spaces. If {A0β1 : β1 ≤ λ1} and {A00β
2 : β2 ≤ λ2} are closed coverings of X and Y , respectively, by finite-dimensional subsets satisfying (1.1), (1.2), then {Aβ : β ≤ λ1⊕ λ2}, where Aβ = S{Aβ1 × Aβ2 : β1⊕ β2 = β, β1 ≤ λ1, β2 ≤ λ2}, is a closed covering of X × Y by finite-dimensional subsets satisfying (1.1), (1.2).
P r o o f. Since dim(X × Y ) ≤ dim X + dim Y for any non-empty compact
or metrizable spaces X and Y (see [2], Theorem 3.2.13 or 4.1.21), our lemma follows from the proof of Theorem 5 in [4].
3. The tools. This section does not directly concern dimension theory, but it contains the main tools of the paper. We shall describe a few topolog- ical operations, introduce a technical notion important for the sequel, and prove their basic—from our point of view—properties.
Let |λ| ≤ τ and let T be an arbitrary set of cardinality τ . Suppose we are given a family X = {(Xα, xα) : α < λ} (a family X = {(Xα, xα, %α) : α < λ}) of pointed compact spaces (of pointed metric spaces). For all α, ξ < λ and every t ∈ T , set (Xα,ξ,t, xα,ξ,t) = (Xα, xα) (respectively, (Xα,ξ,t, xα,ξ,t, %α,ξ,t) = (Xα, xα, %α)).
1. The compact case. We denote by JC(X , τ ) the quotient space obtained from the Alexandrov compactification of the topological sum L{Xα,ξ,t : α, ξ < λ, t ∈ T }—the unique point of the remainder is denoted by xλ—by identifying the set {xα,ξ,t: α, ξ < λ, t ∈ T } ∪ {xλ} to a point O ∈ JC(X , τ );
the spaces Xα,ξ,t can be identified in a natural way with the respective subspaces of JC(X , τ ).
2. The metric case. We denote by JM(X , τ ) the space obtained from L{Xα,ξ,t: α, ξ < λ, t ∈ T } by identifying the set {xα,ξ,t: α, ξ < λ, t ∈ T } to a point O ∈ JM(X , τ )—the spaces Xα,ξ,t can be identified in a natural way with the respective subspaces of JM(X , τ )—equipped with the metric
% defined by letting %(x, y) = %α,ξ,t(x, y) if x, y ∈ Xα,ξ,t for some α, ξ < λ, t ∈ T and %(x, y) = %(x, O) + %(y, O) otherwise.
The reader can easily check that we have obtained a compact space JC(X , τ ) (a metric space (JM(X , τ ), %)).
The equivalence of metrics %α and σα on Xα for every α < λ does not imply that the corresponding metrics % and σ on JM(X , τ ) are equivalent.
In the sequel, we shall consider the topological space JM(X , τ ), where X is a family of pointed metrizable spaces. One should understand that when defining the space (JM(X , τ ), %), we have fixed arbitrary metrics %αcompat- ible with the topologies on Xα (except in the proof of Corollary 4.4, where we shall additionally assume that the metrics %αare complete for completely metrizable spaces Xα), and that the topology on JM(X , τ ) is induced by the metric %.
In order to give a common proof of both our theorems, we shall some- times use the symbol J (X , τ ) instead of JC(X , τ ) while proving the gener- alization of (1.6), and instead of JM(X , τ ) while proving the generalization of (1.5).
Then
J (X , τ ) =[
{Xα,ξ,t: α, ξ < λ, t ∈ T }
and O is the unique common point of every pair of subspaces Xα1,ξ1,t1, Xα2,ξ2,t2.
Let ω(0) = 0 and ω(m) = 1 + 2 + . . . + m for every m ≥ 1. We shall denote by Jω(X , τ ) the subspace of [J (X , τ )]ℵ0 consisting of all points (x1, . . . , xj, . . .) such that
(3.1) if O 6= xj ∈ Xαj,ξj,tj for a j, then xk ∈[
{Xαk,ξk,tk : αk ≤ ξj, ξk < λ, tk ∈ T } for all k 6= j,
(3.2) there exists an m ∈ N such that {j : xj 6= O} ⊆ {ω(m) + 1, . . . , ω(m + 2)}.
The point (O, O, . . .) ∈ Jω(X , τ ) will also be denoted by O.
3.1. Proposition. If the weight of Xα is not greater than τ for every α < λ, then the weight of Jω(X , τ ) is not greater than τ .
The space Jω(X , τ ) is a closed subspace of [J (X , τ )]ℵ0; thus JCω(X , τ ) is compact , and JMω(X , τ ) is metrizable.
If Xα is strongly countable-dimensional for every α < λ, then JCω(X , τ ) is strongly countable-dimensional.
P r o o f. The first part of our proposition follows from the existence of a continuous mapping of the space L{Xα : α < λ} onto J (X , τ ), the inequality |λ| ≤ τ , and the inclusion Jω(X , τ ) ⊆ [J (X , τ )]ℵ0.
If a point (x1, . . . , xj, . . .) ∈ [J (X , τ )]ℵ0 does not satisfy (3.1), then there are distinct j, k = 1, 2, . . . such that O 6= xj ∈ Xαj,ξj,tj and
xk6∈[
{Xαk,ξk,tk : αk ≤ ξj, ξk < λ, tk ∈ T }.
Since J (X , τ ) = S{Xα,ξ,t : α, ξ < λ, t ∈ T } and O ∈ Xα,ξ,t for every α, ξ < λ, t ∈ T , we conclude that
O 6= xk∈ Xαk,ξk,tk for some αk > ξj, ξk < λ, tk ∈ T ; then the set
U = {(y1, . . . , yj, . . .) ∈ [J (X , τ )]ℵ0:
O 6= yj ∈ Xαj,ξj,tj, O 6= yk ∈ Xαk,ξk,tk} is a neighbourhood of (x1, . . . , xj, . . .) and no point of U satisfies (3.1).
Thus the set C of all points of [J (X , τ )]ℵ0 satisfying (3.1) is closed.
The set D of all points of [J (X , τ )]ℵ0 satisfying (3.2) can be represented as the union of the family {Jm : m ∈ N }, where Jm is the subspace of [J (X , τ )]ℵ0 consisting of all points (x1, . . . , xj, . . .) such that
{j : xj 6= O} ⊆ {ω(m) + 1, . . . , ω(m + 2)},
for m ∈ N . Since Jm is closed in [J (X , τ )]ℵ0 for every m ∈ N , the point (O, O, . . .) satisfies (3.2), and every point of [J (X , τ )]ℵ0 distinct from (O, O, . . .) has a neighbourhood U such that U ∩ Jm= ∅ for all but finitely many m ∈ N , so that the set D is closed in [J (X , τ )]ℵ0.
Thus Jω(X, τ ) = C ∩D is a closed subspace of [J (X , τ )]ℵ0and the second part of the proposition is established.
Since JCω(X, τ ) is a closed subspace of D, and D =S{Jm : m ∈ N }, it suffices to show that Jm is a strongly countable-dimensional space for each m ∈ N ; but every Jm is homeomorphic to a product of finitely many copies of JC(X, τ ), so, by Theorem 3.2.13 in [2], it suffices to show that JC(X, τ ) is strongly countable-dimensional.
Let Xα=S{Xα,n : n ∈ N }, where Xα,n is compact and dim Xα,n ≤ n for every n ∈ N ; put Xα,ξ,t,n= Xα,n for every α, ξ < λ, t ∈ T, n ∈ N , and
Xn= {O} ∪[
{Xα,ξ,t,n : α, ξ < λ, t ∈ T }
for n ∈ N . Then Xnis closed subspace of JC(X, τ ) and JC(X, τ ) =S{Xn : n ∈ N }. From the definition of the covering dimension it follows directly that dim Xn ≤ n for every n ∈ N . Hence the third part of the proposition is also established.
Suppose we are given a space X, its subsets F and U , closed and open, respectively, and a space Y , its subspace A, and a point y ∈ A . We say that the sixtuple F , U , X, A, y, Y has property (S) if for every open subset V ⊆ U , there exists a mapping f : X → Y separating points of F ∩ V and closed sets in X such that f−1(y) = X − V and f (F ) ⊆ A.
3.2. Proposition. Let F be a closed subset and U an open subset of a space X. Consider a sequence ∅ = U0 ⊆ U1 ⊆ . . . ⊆ Uk = U of open subsets of X and a sequence (Y1, y1), . . . , (Yk, yk) of pointed spaces and set Fi+1= F − Ui for i = 0, . . . , k − 1 and (Y, y) = (Y1× . . . × Yk, (y1, . . . , yk)).
If the sixtuple Fi, Ui, X, Yi, yi, Yi has property (S) for i = 1, . . . , k, then so does F , U , X, Y , y, Y .
P r o o f. Let V ⊆ U be an open subset; define Vi= V ∩Uifor i = 1, . . . , k.
Since Fi, Ui, X, Yi, yi, Yihas property (S), there exists a mapping fi: X → Yi
separating points of Fi∩ Viand closed sets in X such that fi−1(yi) = X − Vi. Let f = f14 . . . 4fk : X → Y = Y1× . . . × Yk; then
f−1(y) = f1−1(y1) ∩ . . . ∩ fk−1(yk)
= X −
k
[
i=1
Vi= X −
k
[
i=1
(V ∩ Ui) = X − V
and f separates points of (V1∩ F1) ∪ . . . ∪ (Vk∩ Fk) and closed sets in X.
Since
(V1∩ F1) ∪ . . . ∪ (Vk∩ Fk)
= (V ∩ U1∩ (F − U0)) ∪ . . . ∪ (V ∩ Uk∩ (F − Uk−1))
= V ∩ F ∩ [(U1− U0) ∪ . . . ∪ (Uk− Uk−1)] = V ∩ F, the proof is complete.
3.3. Theorem. Let X be a metrizable space of weight τ , F and U its closed and open subset , respectively. Consider an open covering {Uα: α <
λ} of U and a family X = {(Xα, xα) : α < λ} of pointed compact or metrizable spaces. If for every α < λ the sixtuple F , Uα, X, Xα, xα, Xα
has property (S), then so does F , U , X, Jω(X , τ ), O, Jω(X , τ ).
P r o o f. Take an open subset V ⊆ U and put Vα= V ∩ Uα for α < λ.
Let G be a locally finite open refinement of {Vα : α < λ}, and H an open covering each of whose elements intersects only finitely many members of G. Choose a refinement V =S{Vm : m = 1, 2, ...} of the covering {G ∩ H : G ∈ G, H ∈ H} of V , where each family Vm is discrete in X. Let Vm = S{S Vn : n = 1, . . . , m} for m = 1, 2, . . . By Lemma 2.1, there exists an open shrinking {Wm: m = 1, 2, . . .} of {Vm: m = 1, 2, . . .} such that Wl∩Wm= ∅ whenever |m − l| > 1. For j ∈ N , take an m ∈ N and an n = 1, . . . , m + 1 such that j = ω(m) + n, and define Wj = {Wm+1 ∩ V : V ∈ Vn} and W =S{Wj : j = 1, 2, . . .}.
The families defined above have the following properties:
(3.3) the family Wj is discrete in X and its elements are open subsets of V for every j = 1, 2, . . . ,
(3.4) the family W is a covering of V ,
(3.5) for every W ∈ W, there exist α, ξ < λ such that W ⊆ Uα and if W ∩ W06= ∅ for a W0∈ W, then W0⊆ Uα0 for some α0≤ ξ, (3.6) if W ∩ W06= ∅ for some W ∈ Wj and W0∈ Wk, then
j, k ∈ {ω(m) + 1, . . . , ω(m + 2)} for some m ∈ N ; we denote by α(W ) and ξ(W ) the smallest α and ξ satisfying (3.5).
Fix a j ∈ N . Since the weight of X is equal to τ , we have |Wj| ≤ τ , and therefore for our set T of cardinality τ there exists an injection θ : Wj → T . For every W ∈ Wj, there exists, by the assumptions of our theorem, a mapping fW : X → Xα(W ),ξ(W ),θ(W )separating points of W ∩ F and closed sets in X such that fW−1(xα(W ),ξ(W ),θ(W )) = X − W .
It follows from (3.3) that the family of mappings {fW : W ∈ Wj} yields a mapping fj : X → J (X , τ ) separating points of (S Wj) ∩ F and closed sets in X such that fj−1(O) = X − (S Wj).
Let f = 4{fj : j = 1, 2, . . .}; then f−1(O) = X − V by (3.4), and f separates points of F ∩ V and closed sets in X. It follows from (3.5) that every point f (x) ∈ [J (X , τ )]ℵ0 satisfies (3.1), and by (3.6), it also satisfies (3.2). Thus f (X) ⊆ Jω(X , τ ), and the proof of our theorem is complete.
4. The spaces Mα(τ ) and Zα(τ ), their basic properties. For every α such that |α| ≤ τ , we define by induction a metrizable space Mα(τ ) and a compact space Zα(τ ). In this section, we establish the properties of these spaces announced in the abstract, except for the inequalities D(Mα(τ )) ≤ α, D(Zα(τ )) ≤ α that are proved in Section 5. In Section 2 (see Lemma 2.3) and in Section 3, we prepared the tools that will now allow us to carry out in the general case the (suitably modified) argument used by L. Luxemburg in the special case of separable spaces (see [8], the proofs of Theorems 1.3 and 1.4).
We shall distinguish inductively a point O in every Mα(τ ) and Zα(τ ).
Let Mn(τ ) = Kn(τ ) and let Zn(τ ) be an n-dimensional compactification of weight τ of Kn(τ ) (see [2], Theorem 3.3.3) for n = 0, 1, . . .; we have already distinguished the point O ∈ Mn(τ ) ⊆ Zn(τ ).
Let α = λ + n; then λ = λ1+ . . . + λk, where λj is a prime component for j = 1, . . . , k and λ1≥ λ2≥ . . . ≥ λk.
We first define some auxiliary pointed spaces Mλ0(τ ) and Zλ0(τ ). If k = 1, then let Zλ0(τ ) = JCω(X , τ ), where X = {(Zα(τ ), O) : α < λ}, and Mλ0(τ ) = JMω(X , τ ), where X = {(Mα(τ ), O) : α < λ}. If k > 1, then let Zλ0(τ ) = Zλ1(τ ) × . . . × Zλk(τ ) and Mλ0(τ ) = Mλ1(τ ) × . . . × Mλk(τ ). In the first case, we have already distinguished the points O (see the definition of Jω(X , τ ) in Section 3); in the second case, take (O, . . . , O) as O.
Now, let Mα(τ ) = {(x, y) ∈ Mλ0(τ ) × Kn+1(τ ) : if x = O, then y ∈ Kn(τ )}, and Zα(τ ) = Zλ0(τ ) × Zn(τ ); in both cases take (O, O) as O.
Using Proposition 3.1 and induction on α we obtain the following propo- sition.
4.1. Proposition. Let |α| ≤ τ . The space Mα(τ ) is metrizable and the space Zα(τ ) is compact. The space Zα(τ ) is strongly countable-dimensional.
The weight of Zα(τ ) and of Mα(τ ) is equal to τ .
4.2. Theorem. Let X be a metrizable space of weight τ , F and U its subsets, closed and open, respectively. If D(F ∩ U ) ≤ λ + n, where λ is 0 or a limit ordinal , then the sixtuples F , U , X, Mλ+n(τ ), O, Mλ+n+1(τ ) and F , U , X, Zλ+n+1(τ ), O, Zλ+n+1(τ ) have property (S). If D(F ∩ U ) <≤ λ, then the sixtuples F , U , X, Mλ0(τ ), O, Mλ0(τ ) and F , U , X, Zλ0(τ ), O, Zλ0(τ ) have property (S).
P r o o f. We use induction on λ. From Lemma 2.3 it follows that
(4.1) for every metrizable space X of weight τ and its subsets F and U , closed and open, respectively, such that dim(F ∩ U ) ≤ n, the sixtuple F , U , X, Kn(τ ), O, Kn+1(τ ) has property (S).
Thus the theorem holds for λ = 0.
Let λ be a limit ordinal. Since D(F ∩ U ) ≤ λ + n, there exists a closed covering {Aβ : β ≤ λ} of the space F ∩ U by finite-dimensional subsets satisfying (1.1)–(1.4). Let W = U −Aλ; then W is open and D(F ∩W ) <≤ λ.
We are going to prove that
(4.2) the sixtuples F , W , X, Mλ0(τ ), O, Mλ0(τ ) and F , W , X, Zλ0(τ ), O, Zλ0(τ ) have property (S).
C a s e 1: λ is a prime component. Let Wγ = W −S{Aβ : γ ≤ β < λ}
for γ < λ. By (1.1), the sets Wγ are open; obviously, W =S{Wγ : γ < λ}.
If γ is a limit ordinal, then D(F ∩ Wγ) <≤ γ; otherwise it follows from (1.4) that D(F ∩ Wγ) < γ. Hence, by the inductive assumption, the sixtuples F , Wγ, X, Mγ(τ ), O, Mγ(τ ) and F , Wγ, X, Zγ(τ ), O, Zγ(τ ) have property (S) for γ < λ. Thus, by Theorem 3.3, so do F , W , X, Mλ0(τ ), O, Mλ0(τ ) and F , W , X, Zλ0(τ ), O, Zλ0(τ ).
C a s e 2: λ = λ1+ . . . + λk, where k > 1. Let Wi= W −[
{Aβ : λ1+ . . . + λi≤ β ≤ λ}
for i = 1, . . . , k. By (1.1), the sets Wi are open; obviously, ∅ = W0⊆ W1⊆ . . . ⊆ Wk= W . Put Fi+1= F − Wi for i = 0, . . . , k − 1; then
Fi+1∩ Wi+1
=[
{Aβ : λ0+ . . . + λi≤ β ≤ λ}
−[
{Aβ : λ0+ . . . + λi+1≤ β ≤ λ} , where λ0 = 0, and one can easily check that D(Fi+1∩ Wi+1) <≤ γi+1 for i = 0, . . . , k − 1. By the inductive assumption, the sixtuples F , Wi, X, Mλi(τ ), O, Mλi(τ ) and Fi, Wi, X, Zλi(τ ), O, Zλi(τ ) have property (S) for i = 1, . . . , k. Hence, by Proposition 3.2, so do F , W , X, Mλ0(τ ), O, Mλ0(τ ) and F , W , X, Zλ0(τ ), O, Zλ0(τ ).
If D(F ∩ U ) <≤ λ, then Aλ = ∅, and the proof of the first part of our theorem is complete.
If D(F ∩ U ) ≤ λ + n, then dim Aλ≤ n by (1.3), and the second part of our theorem follows from (4.1) and (4.2).
4.3. Corollary. Let |α| ≤ τ . If X is a metrizable space of weight τ such that D(X) ≤ α, then X is homeomorphic to a subspace of Mα(τ ) and to a subspace of Zα+1(τ ).
P r o o f. It suffices to apply Theorem 4.2 to F = U = X.
Denote by τ+ the smallest cardinal greater than τ .
4.4. Corollary. For every cardinal τ , there exist strongly countable- dimensional spaces Zτ and Mτ of weight τ+, compact and completely metriz- able, respectively, such that each strongly countable-dimensional completely metrizable space of weight τ is homeomorphic to a subspace of Zτ and to a subspace of Mτ.
P r o o f. For every n ∈ N the space Kn(τ ) is completely metrizable as a Gδ subset of [J (τ )]ℵ0; the space (JM(X , τ ), %) is complete whenever the spaces (Xα, %α) are complete for α < λ (see the remark following the definition of J (X , τ )), and therefore, by Proposition 3.1, the space JMω(X , τ ) is completely metrizable for every α such that |α| ≤ τ .
Since D(X) < ∆ and |D(X)| ≤ τ for every strongly countable-dimen- sional completely metrizable space X of weight τ , it suffices to take the Alexandroff compactification ofL{Zα(τ ) : |α| ≤ τ } as Zτ, andL{Mα(τ ) :
|α| ≤ τ } as Mτ.
Note that Corollary 4.4 is obvious under GCH.
5. The D-dimension of Mα(τ ) and Zα(τ ). In this section, we evaluate D(Mα(τ )) and D(Zα(τ )) for every α such that |α| ≤ τ .
Let T be our set such that |T | = τ . Fix an l ≥ 1 and a λ such that
|λ| ≤ τ . We denote by J (X , τ )l the subspace of J (X , τ )l consisting of all points (x1, . . . , xj, . . . , xl) satisfying (3.1).
5.1. Lemma. Let λ be a prime component and X = {(Xα, xα) : α < λ} a family of pointed metrizable or compact spaces of weight τ . If D(Xα) ≤ α for every α < λ, then J (X , τ )l has the following property:
(5.1) there exists its closed covering {Aβ : β ≤ λ} by finite-dimensional subsets satisfying (1.1), (1.2), (1.4) and such that Aλ= {O}.
P r o o f. Since J (X , τ ) = S{Xα,ξ,t : α, ξ < λ, t ∈ T } (see Section 3), by (3.1),
J (X , τ )l=[
{Rαξ11,...,ξ,...,αll : α1, ξ1, . . . , αl, ξl < λ and
maxi6=j αi≤ ξj for every j = 1, . . . , l}, where Rξα11,...,ξ,...,αll =S{Xα1,ξ1,t1× . . . × Xα1,ξ1,t1 : t1, . . . , tl∈ T }.
Take a closed covering {Aαβ : β ≤ λ(α)} of Xα by finite-dimensional subsets satisfying (1.1)–(1.4), where λ(α) is the largest limit ordinal less than or equal to α, and put Bβα= Aαβ for β < λ(α), Bαβ = ∅ for λ(α) ≤ β < α, Bαα= Aαλ(α).
Further, let
Aα,ξ,tβ = Bβα for α, ξ < λ, t ∈ T, β ≤ α,
Aα,ξβ = {O} ∪[
{Aα,ξ,tβ : t ∈ T } for α, ξ < λ, β ≤ α;
and Aα,ξβ = ∅ for α, ξ < λ, α < β ≤ λ. Then the family {Aα,ξβ : β ≤ λ} is a closed covering of the subspace Rα,ξ = S{Xα,ξ,t : t ∈ T } of J (X , τ ) by finite-dimensional subsets satisfying (1.1), (1.2), (1.4).
Define Aβ =[
{Aαβ1,ξ1
1 × . . . × Aαβl,ξl
l : β1⊕ . . . ⊕ βl = β, α1, ξ1, . . . , αl, ξl< λ, maxi6=j αi≤ ξj for every j = 1, . . . , l}
for β < λ and let Aλ= {O}.
Obviously, Aβ is a subset of J (X , τ )l for every β ≤ λ. The β = λ is the greatest number β ≤ λ such that O ∈ Aβ. Let O 6= (x1, . . . , xl) ∈ J (X , τ )l ⊆ [J (X , τ )]l. Consider a k = 1, . . . , l. If xk 6= O, then there are a unique αk< λ and a unique ξk< λ such that xk ∈ Rαk,ξk, and a greatest βk ≤ αk
such that xk ∈ Aαβk,ξk
k ; since (x1, . . . , xl) satisfies (3.1), αk ≤ ξi for each k 6= i = 1, . . . , l such that xi 6= O. If xk = O, then put αk = βk = min{ξi: i = 1, . . . , l and O 6= xi∈ Rαi,ξi}, ξk = max{αi: i = 1, . . . , l}.
Since O ∈ Aα,ξβ for every α, ξ < λ and β ≤ α, we have (x1, . . . , xl) ∈ Aαβ11,ξ1 × . . . × Aαβl,ξl
l , and since maxi6=jαi ≤ ξj for every j = 1, . . . , l, it follows that Aαβ11,ξ1× . . . × Aαβl,ξl
l ⊆ Aβ for β = β1⊕ . . . ⊕ βl (we have β < λ, because β ≤ αi< λ for every i = 1, . . . , l and λ is a prime component).
It follows from the definition of Aβ’s that the chosen β is the greatest β such that (x1, . . . , xl) ∈ Aβ. Thus, we have shown that {Aβ : β ≤ λ} is a covering of J (X , τ )l and (1.2) is satisfied.
We shall now prove that Aβ is closed in [J (X , τ )]l for every β < λ (for β = λ this is obvious).
Since the equation β1⊕ . . . ⊕ βl = β has only finitely many solutions β1, . . . , βl, it suffices to prove that the set
Aβ1,...,βl =[
{Aαβ1,ξ1
1 × . . . × Aαβl,ξl
l : α1, ξ1, . . . , αl, ξl < λ and max
i6=j αi≤ ξj for every j = 1, . . . , l}
is closed. Take an (x1, . . . , xl) 6∈ Aβ1,...,βl. If there exists an i ∈ {1, . . . , l}
such that xi6∈ Aαβi,ξi
i for every αi, ξi< λ, then U =
n
(y1, . . . , yl) ∈ [J (X , τ )]l: yi6∈[ {Aαβi,ξi
i : αi, ξi< λ}
o is a neighbourhood of (x1, . . . , xl) and U ∩ Aβ1,...,βl = ∅.
Thus assume that for every i = 1, . . . , l, xi∈ Aαβi,ξi
i for some αi, ξi < λ.
Put αi = βi whenever xi = O, and if xi 6= O, then let αi be the unique ordinal less than λ such that xi∈ Rαi,ξi for some ξi< λ. Further, let ξi be
the unique ordinal less than λ such that xi∈ Rαi,ξi whenever xi 6= O, and let ξi= max{αj : i 6= j = 1, . . . , l} whenever xi= O.
Then xi ∈ Aαβi,ξi
i for i = 1, . . . , l. Since (x1, . . . , xl) 6∈ Aβ1,...,βl, we have ξj < αi for some distinct i, j = 1, . . . , l; this is possible only for j such that xj 6= O. If also xi6= O, then
U = {(y1, . . . , yl) ∈ [J (X , τ )]l : O 6= yi∈ Rαi,ξi, O 6= yj ∈ Rαj,ξj} is a neighbourhood of (x1, . . . , xl) and U ∩ Aβ1,...,βl ⊆ U ∩ J (X , τ )l = ∅. If xi= O, then βi= αi> ξj, so that
U = {(y1, . . . , yl) ∈ [J (X , τ )]l : O 6= yj ∈ Rαj,ξj} is a neighbourhood of (x1, . . . , xl) satisfying U ∩ Aβ1,...,βl = ∅.
Thus Aβ is a closed subset of [J (X , τ )]l for every β ≤ λ.
In order to prove (1.1), observe that [{Aβ : δ ≤ β ≤ λ}
=[n[nh[
{Aαβ1,ξ1
1 : δ1≤ β1≤ λ1}i
× . . . ×h[
{Aαβl,ξl
l : δl ≤ βl ≤ λl}i : α1, ξ1, . . . , αl, ξl< λ, max
i6=j αi< ξj for j = 1, . . . , lo
: δ1⊕ . . . ⊕ δl= δo , because if β = β1⊕ . . . ⊕ βl ≥ δ, then there are δ1 ≤ β1, . . . , δl ≤ βl such that δ1⊕ . . . ⊕ δl= δ.
Since S{Aαβi,ξi
i : δi ≤ βi ≤ αi} is a closed subset of J (X , τ ) for every αi, ξi< λ and δi≤ αi, by the above equality, the proof of the closedness of S{Aβ : δ ≤ β ≤ λ} can be carried out exactly as the proof of the closedness of Aβ.
Let Cβ =[
{Aαβ1,ξ1
1 × . . . × Aαβl,ξl
l : α1, ξ1, . . . , αl, ξl < λ, β = β1⊕ . . . ⊕ βl}.
It follows from Lemma 2.4 that dim Cβ ≤ m for every β = γ + m < λ, where γ is 0 or a limit ordinal. Since Aβ is a closed subset of Cβ, dim Aβ ≤ m. Thus the family {Aβ : β ≤ λ} has the property described in (1.4); in particular, the sets Aβ are finite-dimensional.
5.2. Lemma. Let λ be a prime component such that |λ| ≤ τ and X = {(Xα, xα) : α < λ} a family of pointed metrizable or compact spaces of weight τ . If D(Xα) ≤ α for every α < λ, then Jω(X , τ ) has property (5.1).
P r o o f. For m = 0, 1, . . . , let Jm be the subspace of Jω(χ, τ ) consisting of all points (x1, . . . , xj, . . .) such that {j : xj 6= O} ⊆ {ω(m) + 1, . . . , ω(m + 2)}. Obviously, Jm’s are closed subsets of Jω(X , τ ) and Jω(X , τ ) =S{Jm: m = 0, 1, . . .}, and by Lemma 5.1, Jm’s have property (5.1). Take a covering {Amβ : β ≤ λ} of Jm satisfying (5.1) for m = 0, 1, . . . For β = γ + m, where
γ < λ is 0 or a limit ordinal, put
Aβ = A0γ+m∪ . . . ∪ Amγ+0;
let Aλ= {O}. Then {Aβ : β ≤ λ} is a covering of Jω(X , τ ) satisfying (5.1).
5.3. Theorem. For every α such that |α| ≤ τ , D(Mα(τ )) ≤ α and D(Zα(τ )) ≤ α .
P r o o f. We apply induction on α and we simultaneously prove that for every λ,
(5.2) Zλ0(τ ) and Mλ0(τ ) have property (5.1).
For all finite α, the theorem follows directly from the definition of Mα(τ ) and Zα(τ ). Let α be infinite. If α = λ is a prime component, then (5.2) follows from Lemma 5.2 and the inductive assumption. If α = λ is a limit ordinal, but not a prime component, then (5.2) follows from Lemma 2.4 and the inductive assumption. For α = λ + n, our theorem is an immediate consequence of (5.2) for λ.
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DEPARTMENT OF MATHEMATICS UNIVERSITY OF WARSAW BANACHA 2
00-913 WARSZAWA 59, POLAND
Received 20 September 1990 ; in revised form 13 June 1991
